• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Fluctuations of an interface with hammock potential

+ 11 like - 0 dislike

This question is related to that one. I ask it here since comments are too short for the extended discussion that was going on there.

I am interested in a very simple interface model. To each $x\in\mathbb{Z}^2$, we associate a random height $h_x\in\mathbb{R}$. Let $\Lambda_N=\{-N,\ldots,N\}^2$. Assume $h_x\equiv 0$ outside $\Lambda_N$. To a pair of neighbouring heights, we associate an energy $0$ if $|h_x-h_y| < 1$ and an energy $+\infty$ otherwise. We then consider the corresponding Gibbs measure. In other words, we put the uniform measure on height configurations satisfying $|h_x-h_y| < 1$ for all pairs of neighbouring vertices, and equal to $0$ outside $\Lambda_N$.

It is an open problem to prove that the variance of $h_0$ diverges as $\log N$, as $N\to\infty$ (actually, it's even open to prove that it diverges at all!).

On the other hand, it is known to hold, if one replaces $+\infty$ by a suitable function diverging outside the interval (fast enough to guarantee existence of the measure, of course). Obviously, one cannot take the limit in the known arguments...

My question: What are quantitative heuristic arguments implying such a claim. By quantitative, I mean that I don't want something like "by analogy with the discrete massless free field", because I already know that ;) . I'd really like a non-rigorous, but mathematical derivation.

Update (April 27, 2014): two colleagues have been able to (rigorously) settle this question in a slightly different geometry (periodic boundary conditions, the spin at the origin forced to be $0$). Their preprint can be found here: arXiv:1404.5895. Nevertheless, I'm still intertested in good physical heuristics.

This post has been migrated from (A51.SE)

asked Oct 7, 2011 in Theoretical Physics by Yvan Velenik (1,110 points) [ revision history ]
edited Apr 27, 2014 by Yvan Velenik
Thanks for sharing this. I would like to ask if the temperature is relevant in this problem. I mean if I would like to try an argument using cluster expansion for the second moment, in the regime of low temperatures it could this have any relevance for the problem ?

This post has been migrated from (A51.SE)
No, temperature should be irrelevant (energy being always $0$ or $\infty$). I (and quite a few others) have tried various approaches to this problem, but it still resists ;) . Actually, there are very few tools to deal with such systems of purely entropic nature (and many interesting such systems).

This post has been migrated from (A51.SE)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights